Self-Adjointness, Symmetries, and Conservation Laws for a Class of ...

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 407908, 6 pages http://dx.doi.org/10.1155/2013/407908

Research Article Self-Adjointness, Symmetries, and Conservation Laws for a Class of Wave Equations Incorporating Dissipation Yang Wang and Long Wei Institute of Applied Mathematics and Engineering Computations, Hangzhou Dianzi University, Zhejiang 310018, China Correspondence should be addressed to Long Wei; [email protected] Received 17 February 2013; Revised 30 April 2013; Accepted 13 May 2013 Academic Editor: Qian Guo Copyright © 2013 Y. Wang and L. Wei. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this work, we study the nonlinear self-adjointness and conservation laws for a class of wave equations with a dissipative source. We show that the equations are nonlinear self-adjoint. As a result, from the general theorem on conservation laws proved by Ibragimov and the symmetry generators, we find some conservation laws for this kind of equations.

1. Introduction There have been abundant literatures that have contributed to the studies of Lie symmetry classification of various classes of (1 + 1)-dimensional nonlinear wave equations and their individual members. Probably, Barone et al. [1] were the first to study the nonlinear wave equation 𝑢𝑡𝑡 = 𝑢𝑥𝑥 + 𝐹(𝑢) by means of symmetry method. Motivated by a number of physical problems, Ames et al. [2] investigated group properties of quasilinear 𝑢𝑡𝑡 = [𝑓(𝑢)𝑢𝑥 ]𝑥 . Later, Torrisi and Valenti [3, 4], generalizing above equations, have investigated the symmetries of the following equation: 𝑢𝑡𝑡 = [𝑓 (𝑢) 𝑢𝑥 + 𝑔 (𝑥, 𝑢)]𝑥 .

(1)

Furthermore, classification results for the equation 𝑢𝑡𝑡 + 𝐾(𝑢)𝑢𝑡 = [𝐹(𝑢)𝑢𝑥 ]𝑥 can be found in [5, 6]. An expanded form of the latter equation 𝑢𝑡𝑡 + 𝐾 (𝑢) 𝑢𝑡 = [𝐹(𝑢)𝑢𝑥 ]𝑥 + 𝐻 (𝑢) 𝑢𝑥

(2)

was studied by Kingston and Sophocleous [7]. In the papers mentioned above, many interesting results including Lie point and nonlocal symmetries classification were systematically investigated. In this paper, we consider a subclass of (2) with 𝐾(𝑢) = 𝛼: 𝑢𝑡𝑡 + 𝛼𝑢𝑡 − [𝑓 (𝑢) 𝑢𝑥 + 𝑔 (𝑢)]𝑥 = 0,

functions, 𝑡 is the time coordinate, 𝑥 is the one-space coordinate, and 𝛼 is a nonzero constant. Here, we will focus on the nonlinear self-adjointness and conservation laws for (3). For (3), we cannot find easily the variational structure so it is inconvenient to apply the Noether theorem to construct conservation laws straightforward for this equation. However, it is fortunate that Ibragimov recently proved a result on conservation laws [8], which does not require the existence of a Lagrangian. The Ibragimov theorem on conservation laws provides an elegant way to establish local conservation laws for the equations under consideration. Since the seminal work of Ibragimov [8], more and more works are dedicated to studying the self-adjointness and conservation laws for some equations in mathematical physics and there are many new developments, including strict selfadjointness [9–11], quasi-self-adjointness [12–15], weak selfadjointness [16, 17], and nonlinear self-adjointness [18–21] and some results which have been communicated in the recent literature; for more references, see [21] and references therein. For the sake of completeness, we briefly present the notations, definition of nonlinear self-adjointness, and Ibragimov’s theorem on conservation laws. Let 𝑥 = (𝑥1 , 𝑥2 , . . . , 𝑥𝑛 ) be 𝑛 independent variables, 𝑢 = 𝑢(𝑥) a dependent variable,

(3)

which is also viewed as a special case of (1) with an additional dissipation, where 𝑓(𝑢) and 𝑔(𝑢) are arbitrary differentiable

𝑋 = 𝜉𝑖

𝜕 𝜕 +𝜂 𝜕𝑥𝑖 𝜕𝑢

(4)

2

Abstract and Applied Analysis

a symmetry of an equation 𝐹 (𝑥, 𝑢(1) , 𝑢(2) , . . . , 𝑢(𝑠) ) = 0, 𝐹∗ (𝑥, 𝑢(1) , V(1) , . . . , 𝑢(𝑠) , V(𝑠) ) :=

𝛿L =0 𝛿𝑢

(5) (6)

the adjoint equation to (5), where L = V𝐹 is called formal Lagrangian, V = V(𝑥) is a new dependent variable, and 𝑠 𝜕 𝜕 𝛿 = + ∑ (−1)𝑚 𝐷𝑖1 ⋅ ⋅ ⋅ 𝐷𝑖𝑚 𝛿𝑢 𝜕𝑢 𝑚=1 𝜕𝑢𝑖1 ⋅⋅⋅𝑖𝑚

(7)

Definition 1. Equation (5) is said to be nonlinearly self-adjoint if the equation obtained from the adjoint equation (6) by the substitution V = 𝜙(𝑥, 𝑢) with a certain function 𝜙(𝑥, 𝑢) ≠ 0 is identical with the original equation (5); in other words, the following equations hold: (8)

for some differential function 𝜆 = 𝜆(𝑥, 𝑢, 𝑢(1) , . . .). Particularly, if (8) holds for a certain function 𝜙 such that 𝜙𝑢 ≠ 0 and 𝜙𝑥𝑖 ≠ 0 for some 𝑥𝑖 , (5) is called weak self-adjoint; if (8) holds for a certain function 𝜙 such that 𝜙 = 𝜙(𝑢) ≠ 𝑢 and 𝜙󸀠 (𝑢) ≠ 0, then (5) is called quasi-self-adjoint; if (8) holds for 𝜙 = 𝑢, then (5) is called (strictly) self-adjoint. In the following, we recall the “new conservation theorem” given by Ibragimov in [8]. We will find conservation laws for (3) by this theorem. Theorem 2. Any Lie point, Lie-Bäcklund, and nonlocal symmetry 𝑋 = 𝜉𝑖

𝜕 𝜕 +𝜂 𝜕𝑥𝑖 𝜕𝑢

𝜕L 𝜕L 𝜕L − 𝐷𝑗 ( ) + 𝐷𝑘 𝐷𝑗 ( ) 𝜕𝑢𝑖 𝜕𝑢𝑖𝑗 𝜕𝑢𝑖𝑗𝑘 −𝐷𝑙 𝐷𝑘 𝐷𝑗 (

𝜕L ) + ⋅⋅⋅] 𝜕𝑢𝑖𝑗𝑘𝑙

𝜕L 𝜕L + 𝐷𝑗 (𝑊) [ − 𝐷𝑘 ( ) + ⋅⋅⋅] 𝜕𝑢𝑖𝑗 𝜕𝑢𝑖𝑗𝑘 + 𝐷𝑘 𝐷𝑗 (𝑊) [

In this section, we determine the nonlinearly self-adjoint subclasses of (3). Let 𝐹 = 𝑢𝑡𝑡 + 𝛼𝑢𝑡 − 𝑓𝑢 (𝑢) 𝑢𝑥2 − 𝑓 (𝑢) 𝑢𝑥𝑥 − 𝑔𝑢 (𝑢) 𝑢𝑥 ; L = V (𝑢𝑡𝑡 + 𝛼𝑢𝑡 − 𝑓𝑢 𝑢𝑥2 − 𝑓𝑢𝑥𝑥 − 𝑔𝑢 𝑢𝑥 ) .

(11)

(12)

Computing the variational derivative of this formal Lagrangian 𝛿L 𝜕L 𝜕L 𝜕L 𝜕L 𝜕L − 𝐷𝑥 + 𝐷𝑡2 + 𝐷𝑥2 , (13) = − 𝐷𝑡 𝛿𝑢 𝜕𝑢 𝜕𝑢𝑡 𝜕𝑢𝑥 𝜕𝑢𝑡𝑡 𝜕𝑢𝑥𝑥 we obtain the adjoint equation of (3): 𝐹∗ = −𝛼V𝑡 + V𝑡𝑡 − 𝑓V𝑥𝑥 + 𝑔𝑢 V𝑥 = 0.

(14)

Assume that 𝐹∗ |V=ℎ(𝑡,𝑥,𝑢) = 𝜆𝐹, for a certain function 𝜆, where 𝐹 is given by (11); then we have ℎ𝑢 𝑢𝑡𝑡 + ℎ𝑢𝑢 𝑢𝑡2 + (2ℎ𝑡𝑢 − 𝛼ℎ𝑢 ) 𝑢𝑡 − 𝑓ℎ𝑢 𝑢𝑥𝑥 − 𝑓ℎ𝑢𝑢 𝑢𝑥2 + (𝑔𝑢 ℎ𝑢 − 2𝑓ℎ𝑥𝑢 ) 𝑢𝑥 + ℎ𝑡𝑡 − 𝛼𝑢𝑡 − 𝑓ℎ𝑥𝑥 + 𝑔𝑢 ℎ𝑥 (15) = 𝜆 (𝑢𝑡𝑡 + 𝛼𝑢𝑡 − 𝑓𝑢 𝑢𝑥2 − 𝑓𝑢𝑥𝑥 − 𝑔𝑢 𝑢𝑥 ) . The comparison of the coefficients of 𝑢𝑡𝑡 , 𝑢𝑡2 , 𝑢𝑥2 , 𝑢𝑡 , and 𝑢𝑥 in both sides of (15) yields 𝜆 = ℎ𝑢 ,

(16)

ℎ𝑢𝑢 = 0,

(17)

𝜆𝑓𝑢 = 𝑓ℎ𝑢𝑢 ,

(18)

𝜆𝛼 = 2ℎ𝑡𝑢 − 𝛼ℎ𝑢 ,

(19)

𝜆𝑔𝑢 = 2𝑓ℎ𝑥𝑢 + 𝑔𝑢 ℎ𝑢 ;

(20)

(9)

of (5) provides a conservation law 𝐷𝑖 (𝐶𝑖 ) = 0 for the system comprising (5) and its adjoint equation (6). The conserved vector is given by 𝐶𝑖 = 𝜉𝑖 L + 𝑊 [

2. Nonlinear Self-Adjointness of (3)

then we have the following formal Lagrangian for (3):

denotes the Euler-Lagrange operator. Now, let us state the definition of nonlinear self-adjointness for a equation.

𝐹∗ |V=𝜙 = 𝜆 (𝑥, 𝑢, 𝑢(1) , . . .) 𝐹

The paper is organized as the follows. In Section 2, we discuss the nonlinear self-adjointness of (3). In Section 3, we establish conservation laws for some particular cases of (3) using Theorem 2.

thus, (15) is reduced as ℎ𝑡𝑡 − 𝛼ℎ𝑡 − 𝑓ℎ𝑥𝑥 + 𝑔𝑢 ℎ𝑥 = 0.

(21)

It follows from (17) and (18) that 𝜆𝑓𝑢 = 0.

(10)

𝜕L 𝜕L − 𝐷𝑙 ( ) + ⋅⋅⋅] + ⋅⋅⋅ , 𝜕𝑢𝑖𝑗𝑘 𝜕𝑢𝑖𝑗𝑘𝑙

where 𝑊 = 𝜂 − 𝜉𝑗 𝑢𝑗 is the Lie characteristic function and L = V𝐹 is the formal Lagrangian.

(22)

Equation (22) splits into the following two cases. Case 1 (𝜆 = 0). In this case, from (16) we get that ℎ𝑢 = 0; thus, (16)–(20) are all satisfied. Noting that ℎ does not depend on the function 𝑢, 𝑓 and 𝑔 are two functions of 𝑢, so from (21) we obtain that ℎ𝑡𝑡 − 𝛼ℎ𝑡 = 0,

−𝑓ℎ𝑥𝑥 + 𝑔𝑢 ℎ𝑥 = 0.

(23)


3 Table 2: Symmetries of (3) for some special choices of 𝑓 and 𝑔.

Table 1: Nonlinear self-adjointness of (3). 𝑓 ∀(𝑓𝑢 ≠ 0) ∀(𝑓𝑢 ≠ 0)

𝑔

V

∀ 𝐶5

∀(𝑓𝑢 ≠ 0) 𝐶5 ∫ 𝑓(𝑢)𝑑𝑢 𝐶0 ( ≠ 0)

𝐶1 + 𝐶2 𝑢

𝐶1 + 𝐶2 𝑒𝛼𝑡 (𝐶1 + 𝐶2 𝑒𝛼𝑡 )(𝐶3 + 𝐶4 𝑥) 𝐶1 + 𝐶2 𝑒𝛼𝑡 +(𝐶3 + 𝐶4 𝑒𝛼𝑡 )𝑒𝑥/𝐶5 𝐶1 𝑒(𝐶2 /𝐶0 )𝑥+𝛼𝑡 𝑢 + 𝑏(𝑡, 𝑥)

Selfadjointness Nonlinear Nonlinear

𝑔

𝑓 𝑢

−2

Symmetries 𝜕 𝜕 𝑋3 = 𝑥 −𝑢 𝜕𝑥 𝜕𝑢 𝜕 2𝑢 𝜕 𝑋3 = 𝑥 + 𝜕𝑥 𝑛 𝜕𝑢 𝜕 𝑋3 = 𝑢 , 𝜕𝑢 𝜕 𝜕 1 𝜕 𝑋4 = 𝑡 + 𝑥 − (𝑡 + 𝛼𝑥)𝑢 𝜕𝑥 𝜕𝑡 2 𝜕𝑢 𝜕 𝜕 𝜕 1 𝜕 𝑋3 = 𝑢 , 𝑋4 = 𝑡 + 𝑥 − 𝛼𝑥𝑢 𝜕𝑢 𝜕𝑥 𝜕𝑡 2 𝜕𝑢

ln 𝑢

𝑢𝑛 (𝑛 ≠ 0)

1

1

𝑢

1

1

Nonlinear Weak

𝛼 is a nonzero constant; 𝑏(𝑡, 𝑥) satisfies (27).

Equation (23) can be satisfied by taking some appropriate function ℎ = ℎ(𝑡, 𝑥). Therefore, we can take V = ℎ(𝑥, 𝑡) ≠ 0 such that 𝐹∗ |V=ℎ(𝑡,𝑥,𝑢) = 𝜆𝐹 holds. Thus, (3) is nonlinear selfadjoint in this case. Case 2 (𝑓𝑢 = 0). That is, 𝑓 is a constant with respect to 𝑢; without loss of generality, we set 𝑓 = 𝐶0 ≠ 0. From (17), we assume that ℎ (𝑡, 𝑥, 𝑢) = 𝑎 (𝑡, 𝑥) 𝑢 + 𝑏 (𝑡, 𝑥) .

𝐶0 𝑘󸀠 (𝑥) , 𝑘 (𝑥)

(25)

which implies that 𝑔(𝑢) = 𝐶2 𝑢 + 𝐶1 is a linear function of 𝑢 and 𝑘(𝑥) = 𝐶1 𝑒(𝐶2 /𝐶0 )𝑥 , where 𝐶2 and 𝐶1 are constants. Hence, we have ℎ (𝑡, 𝑥, 𝑢) = 𝐶1 𝑒(𝐶2 /𝐶0 )𝑥+𝛼𝑡 𝑢 + 𝑏 (𝑡, 𝑥) .

𝐷𝑡 (𝐶𝑡 ) + 𝐷𝑥 (𝐶𝑥 ) = 0.

3.1. Lie Symmetries. Now, let us show the Lie classical symmetries for (3). Based on Lie group theory [22], we assume that a Lie point symmetry of (3) is a vector field

𝑋(2) = 𝜉

3. Symmetries and Conservation Laws In this section, we will apply Theorem 2 to construct some conservation laws for (3). First, we show the Lie classical

𝜕 𝜕 𝜕 𝜕 +𝜙 +𝜂 + 𝜂𝑥(1) 𝜕𝑥 𝜕𝑡 𝜕𝑢 𝜕𝑢𝑥

+

(27)

Theorem 3. Let 𝛼 be a nonzero constant, 𝑓(𝑢) an arbitrary function of 𝑢, and the adjoint equation of (3) given by (14); then (3) is nonlinear self-adjointness with the substitution V and function 𝑔(𝑢) given in Table 1.

(29)

on R+ × R × R such that 𝑋(2) 𝐹 = 0 when 𝐹 = 0, and taking into account (3), the operator 𝑋(2) is given as follows:

(26)

which is easy to be solved. In this case, 𝑓(𝑢) = 𝐶0 is a constant, 𝑔(𝑢) = 𝐶2 𝑢 + 𝐶1 , and 𝜆 = ℎ𝑢 = 𝐶1 𝑒(𝐶2 /𝐶0 )𝑥+𝛼𝑡 ; therefore, (3) is a weak self-adjoint. Here, we omit the tedious calculations to obtain the solutions of (23) and (27). In Table 1, we summarize the classification of nonlinear self-adjointness of (3) with the conditions that 𝑓(𝑢) and 𝑔(𝑢) should satisfy. In what follows, the symbol for all means that the corresponding function has no restrictions and 𝑐𝑖 (1 ≤ 𝑖 ≤ 5) are arbitrary constants. Thus, we have demonstrated the following statement.

𝜕 𝜕 𝜕 + 𝜙 (𝑡, 𝑥, 𝑢) + 𝜂 (𝑡, 𝑥, 𝑢) 𝜕𝑥 𝜕𝑡 𝜕𝑢

𝑋 = 𝜉 (𝑡, 𝑥, 𝑢)

Substituting (26) into (22) and using the original equation (3), we derive that 𝑏(𝑡, 𝑥) satisfies 𝑏𝑡𝑡 − 𝛼𝑏𝑡 − 𝐶0 𝑏𝑥𝑥 + 𝐶2 𝑏𝑥 = 0,

(28)

(24)

Taking into account (16) and (19), we have 𝑎𝑡 (𝑡, 𝑥) = 𝛼𝑎(𝑡, 𝑥); thus, 𝑎(𝑡, 𝑥) = 𝑘(𝑥)𝑒𝛼𝑡 . From (20), we deduce that 𝑔𝑢 (𝑢) =

symmetries for (3) for some special choice of 𝑓 and 𝑔. Then applying formula in Theorem 2 to the formal Lagrangian (12), and to the symmetries 𝑋𝑖 and eliminating V by the substitution V given in Table 1, we obtain the conservation law

𝜂𝑡(1)

𝜕 𝜕 (2) 𝜕 + 𝜂𝑥𝑥 + 𝜂𝑡𝑡(2) , 𝜕𝑢𝑡 𝜕𝑢𝑥𝑥 𝜕𝑢𝑡𝑡

(30)

where 𝜂𝑥(1) = 𝐷𝑥 𝜂 − (𝐷𝑥 𝜉) 𝑢𝑥 − (𝐷𝑥 𝜙) 𝑢𝑡 , 𝜂𝑡(1) = 𝐷𝑡 𝜂 − (𝐷𝑡 𝜉) 𝑢𝑥 − (𝐷𝑡 𝜙) 𝑢𝑡 , (2) = 𝐷𝑥 (𝜂𝑥(1) ) − (𝐷𝑥 𝜉) 𝑢𝑥𝑥 − (𝐷𝑥 𝜙) 𝑢𝑥𝑡 , 𝜂𝑥𝑥

(31)

𝜂𝑡𝑡(2) = 𝐷𝑡 (𝜂𝑡(1) ) − (𝐷𝑡 𝜉) 𝑢𝑥𝑡 − (𝐷𝑡 𝜙) 𝑢𝑡𝑡 . The condition 𝑋(2) 𝐹 = 0, when 𝐹 = 0, will yield determining equations. Solving these determining equations, we can obtain the symmetries of (3). For 𝑓(𝑢) and 𝑔(𝑢) arbitrary, the symmetries that are admitted by (3) are 𝑋1 =

𝜕 , 𝜕𝑥

𝑋2 =

𝜕 . 𝜕𝑡

(32)

For some special choices of the functions 𝑓(𝑢) and 𝑔(𝑢), here we omit the details of routine calculations and present the symmetries (besides 𝑋1 and 𝑋2 ) that are admitted by (3) in Table 2.

4

Abstract and Applied Analysis Table 3: Conservation laws of (3) for some special choices of 𝑓 and 𝑔 (𝑛 ≠ 0).

𝑓

𝑔

1 𝑢2

ln 𝑢

𝑢𝑛

Conserved vector with substitution V 𝐶𝑡 = (V𝑡 − 𝛼V)(𝑢 + 𝑥𝑢𝑥 ) − V(𝑢𝑡 + 𝑥𝑢𝑥𝑡 )

Symmetries 𝑥

𝑥

1

𝜕 𝜕 −𝑢 𝜕𝑥 𝜕𝑢

𝐶𝑥 = 𝑥V𝑢𝑡𝑡 + 𝛼𝑥V𝑢𝑡 + V V = 𝐶1 + 𝐶2 𝑒𝛼𝑡 2 2 𝐶𝑡 = ( 𝑢 − 𝑥𝑢𝑥 ) (𝛼V − V𝑡 ) + V ( 𝑢𝑡 − 𝑥𝑢𝑥𝑡 ) 𝑛 𝑛 2 2 𝐶𝑥 = 𝑥 (V𝑢𝑡𝑡 + 𝛼V𝑢𝑡 − 𝑢𝑛 𝑢𝑥 V𝑥 ) + 𝑢𝑛+1 V𝑥 − (1 + ) V𝑢𝑛 𝑢𝑥 𝑛 𝑛 V = (𝐶1 + 𝐶2 𝑒𝛼𝑡 ) (𝐶3 + 𝐶4 𝑥)

𝜕 2𝑢 𝜕 + 𝜕𝑥 𝑛 𝜕𝑢

1 1 (𝑡 + 𝛼𝑥) 𝑢𝑡 + (1 + 𝛼𝑡 + 𝛼2 𝑥) 𝑢 2 2 1 + 𝑡𝑢𝑥𝑡 + 𝑥𝑢𝑥𝑥 ] + V𝑡 [ (𝑡 + 𝛼𝑥) 𝑢 + 𝑥𝑢𝑡 + 𝑡𝑢𝑥 ] 2 1 1 = V [(1 + 𝑥 + 𝛼𝑡)𝑢𝑡 + (𝑡 + 𝛼𝑥)𝑢𝑥 + (𝑡 + 𝛼 + 𝛼𝑥)𝑢 2 2 1 + 𝑡𝑢𝑡𝑡 + 𝑥𝑢𝑥𝑡 ] − V𝑥 [ (𝑡 + 𝛼𝑥) 𝑢 + 𝑥𝑢𝑡 + 𝑡𝑢𝑥 ] 2

𝐶𝑡 = −V [(1 + 𝑥 + 𝛼𝑡) 𝑢𝑥 + 1

𝑢

𝑡

𝜕 𝜕 (𝑡 + 𝛼𝑥)𝑢 𝜕 +𝑥 − 𝜕𝑥 𝜕𝑡 2 𝜕𝑢 𝐶𝑥

V = 𝐶1 + 𝐶2 𝑒𝛼𝑡 + (𝐶3 + 𝐶4 𝑒𝛼𝑡 )𝑒𝑥 or 𝑏(𝑡, 𝑥)

1

1

𝑡

𝛼 𝛼2 𝐶𝑡 = −V ( 𝑥𝑢𝑡 + 𝑥𝑢𝑥𝑥 + 𝑥𝑢 + 𝛼𝑡𝑢𝑥 + 𝑢𝑥 + 𝑡𝑢𝑥𝑡 ) 2 2 𝛼 +V𝑡 (𝑥𝑢𝑡 + 𝑡𝑢𝑥 + 𝑥𝑢) 2 𝛼 𝛼 𝑥 𝐶 = V ( 𝑥𝑢𝑥 + 𝑡𝑢𝑡𝑡 + 𝑢 + 𝛼𝑡𝑢𝑡 + 𝑢𝑡 + 𝑥𝑢𝑥𝑡 ) 2 2 𝛼 −V𝑥 (𝑥𝑢𝑡 + 𝑡𝑢𝑥 + 𝑥𝑢) 2

𝜕 𝜕 𝛼𝑥𝑢 𝜕 +𝑥 − 𝜕𝑥 𝜕𝑡 2 𝜕𝑢

V = (𝐶1 + 𝐶2 𝑒𝛼𝑡 )(𝐶3 + 𝐶4 𝑥) or 𝐶1 𝑒𝛼𝑡 𝑢 + 𝑏(𝑡, 𝑥)

3.2. Conservation Laws. For the symmetries 𝑋, from the formula (10) in Theorem 2, we can obtain readily some conservation laws for (3). For example, we take 𝑓(𝑢) = 𝑢−2 and 𝑔(𝑢) = ln 𝑢; let us construct the conserved vector corresponding to the time translation group with the generator 𝑋2 =

𝜕 . 𝜕𝑡

(33)

For this operator, we have 𝜉𝑡 = 1, 𝜉𝑥 = 0, 𝜂 = 0, and 𝑊 = −𝑢𝑡 . In this case, (3) is nonlinear self-adjoint and becomes 𝐹 = 𝑢𝑡𝑡 + 𝛼𝑢𝑡 + 2𝑢−3 𝑢𝑥2 − 𝑢−2 𝑢𝑥𝑥 − 𝑢−1 𝑢𝑥 .

(34)

The formal Lagrangian is L = V (𝑢𝑡𝑡 + 𝛼𝑢𝑡 + 2𝑢−3 𝑢𝑥2 − 𝑢−2 𝑢𝑥𝑥 − 𝑢−1 𝑢𝑥 ) ,

(35)

and the adjoint equation of (34) is L∗ = V𝑡𝑡 − 𝛼V𝑡 − 𝑢−2 V𝑥𝑥 + 𝑢−1 V𝑥 .

Therefore, we obtain the following conserved vector: 𝐶𝑡 = L + 𝑊 (

𝜕L 𝜕L 𝜕L − 𝐷𝑡 ) + (𝐷𝑡 𝑊) 𝜕𝑢𝑡 𝜕𝑢𝑡𝑡 𝜕𝑢𝑡𝑡

= V (𝑢𝑡𝑡 + 𝛼𝑢𝑡 + 2𝑢−3 𝑢𝑥2 − 𝑢−2 𝑢𝑥𝑥 − 𝑢−1 𝑢𝑥 ) − 𝑢𝑡 (𝛼V − V𝑡 ) − 𝑢𝑡𝑡 V = 2V𝑢𝑥2 𝑢−3 − V𝑢𝑥𝑥 𝑢−2 − V𝑢𝑥 𝑢−1 + 𝑢𝑡 V𝑡 , 𝐶𝑥 = 𝑊 (

𝜕L 𝜕L 𝜕L − 𝐷𝑥 ) + (𝐷𝑥 𝑊) 𝜕𝑢𝑥 𝜕𝑢𝑥𝑥 𝜕𝑢𝑥𝑥

= 𝑢−1 𝑢𝑡 (V − 2𝑢−2 V𝑢𝑥 − 𝑢−1 V𝑥 ) + 𝑢−2 V𝑢𝑥𝑡 , where V satisfies (36). The reckoning shows that the vector (37) satisfies the conservation equation (28). In this case, since (3) is nonlinear self-adjoint, from Table 1 we take V = 𝐶1 + 𝐶2 𝑒𝛼𝑡 . So, the conserved vector is simplified as follows: 𝐶𝑡 = 2V𝑢𝑥2 𝑢−3 − V𝑢𝑥𝑥 𝑢−2 − V𝑢𝑥 𝑢−1 + 𝑢𝑡 V𝑡 ,

(36)

(37)

𝐶𝑥 = 𝑢−1 𝑢𝑡 V (1 − 2𝑢−2 𝑢𝑥 ) + 𝑢−2 V𝑢𝑥𝑡 .

(38)


5

Particularly, if setting V = 1, we have 𝑡

𝐶 = −𝑢𝑥 𝑢

−1

−2

− 𝐷𝑥 (𝑢 𝑢𝑥 ) ,

𝑥

−1

−2

𝐶 = 𝑢 𝑢𝑡 + 𝐷𝑡 (𝑢 𝑢𝑥 ) . (39)

Thus, the conserved vector can be reduced to the form 𝐶𝑡 = −𝑢𝑥 𝑢−1 ,

𝐶𝑥 = 𝑢−1 𝑢𝑡 .

(40)

If setting V = 𝑒𝛼𝑡 , then the conserved vector (37) is simplified as 𝐶𝑡 = 𝛼𝑒𝛼𝑡 𝑢𝑡 − 𝐷𝑥 [𝑒𝛼𝑡 (𝑢−2 𝑢𝑥 + ln 𝑢)] , 𝐶𝑥 = −𝛼𝑒𝛼𝑡 (𝑢−2 𝑢𝑥 + ln 𝑢) + 𝐷𝑡 [𝑒𝛼𝑡 (𝑢−2 𝑢𝑥 + ln 𝑢)] ;

(41)

therefore, it can be reduced as 𝐶𝑡 = 𝛼𝑒𝛼𝑡 𝑢𝑡 ,

𝐶𝑥 = −𝛼𝑒𝛼𝑡 (𝑢−2 𝑢𝑥 + ln 𝑢) .

(42)

In what follows, we omit the tedious calculations and list only the conservation laws of (3) for some special choices of functions 𝑓(𝑢) and 𝑔(𝑢) in Table 3. In Table 3, the function 𝑏(𝑡, 𝑥) satisfies (27), V and the symmetry are taken from Tables 1 and 2, respectively. The reckoning shows that the vector listed in Table 3 satisfies the conservation equation (28) with corresponding substitution V. In the same way above, we can simplify the conserved vector using corresponding substitution V.

4. Conclusions Recently, the new outstanding concepts of nonlinear selfadjoint equations, containing quasi-self-adjoint and weak self-adjoint equations, which extend the self-adjointness to a more generalized meaning, have been introduced in order to find formal Lagrangians of differential equations without variational structure. Using these concepts and the general theorem on conservation laws that is, developed recently [8], nonlinear self-adjointness and conservation laws for (3) for different classes of 𝑓(𝑢) and 𝑔(𝑢) have been discussed. These conservation laws may be useful in mathematical analysis as they provide basic conserved quantity for obtaining various estimates for smooth solutions and defining suitable norms for weak solutions. Furthermore, it could make the construction of the bi-Hamiltonian form easier.

Acknowledgments The authors would like to express their sincere gratitude to the referees for many valuable comments and suggestions which served to improve the paper. The first author is partly supported by NSFC under Grant no. 11101111. The second author is partly supported by Zhejiang Provincial Natural Science Foundation of China under Grant no. LY12A01003 and subjects research and development foundation of Hangzhou Dianzi University under Grant no. ZX100204004-6.

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[2] W. F. Ames, R. J. Lohner, and E. Adams, “Group properties of 𝑢𝑡𝑡 = [𝑓(𝑢)𝑢𝑥 ]𝑥 ,” International Journal of Non-Linear Mechanics, vol. 16, no. 5-6, pp. 439–447, 1981. [3] M. Torrisi and A. Valenti, “Group properties and invariant solutions for infinitesimal transformations of a nonlinear wave equation,” International Journal of Non-Linear Mechanics, vol. 20, no. 3, pp. 135–144, 1985. [4] M. Torrisi and A. Valenti, “Group analysis and some solutions of a nonlinear wave equation,” Atti del Seminario Matematico e Fisico dell’Università di Modena, vol. 38, no. 2, pp. 445–458, 1990. [5] N. H. Ibragimov, Lie Group Analysis of Differential EquationsSymmetries, Exact Solutions and Conservation Laws, CRC, Boca Raton, Fla, USA, 1994. [6] A. Oron and P. Rosenau, “Some symmetries of the nonlinear heat and wave equations,” Physics Letters A, vol. 118, no. 4, pp. 172–176, 1986. [7] J. G. Kingston and C. Sophocleous, “Symmetries and formpreserving transformations of one-dimensional wave equations with dissipation,” International Journal of Non-Linear Mechanics, vol. 36, no. 6, pp. 987–997, 2001. [8] N. H. Ibragimov, “A new conservation theorem,” Journal of Mathematical Analysis and Applications, vol. 333, no. 1, pp. 311– 328, 2007. [9] M. S. Bruzón, M. L. Gandarias, and N. H. Ibragimov, “Selfadjoint sub-classes of generalized thin film equations,” Journal of Mathematical Analysis and Applications, vol. 357, no. 1, pp. 307–313, 2009. [10] I. L. Freire, “Conservation laws for self-adjoint first-order evolution equation,” Journal of Nonlinear Mathematical Physics, vol. 18, no. 2, pp. 279–290, 2011. [11] I. L. Freire, “Self-adjoint sub-classes of third and fourth-order evolution equations,” Applied Mathematics and Computation, vol. 217, no. 22, pp. 9467–9473, 2011. [12] N. H. Ibragimov, “Quasi-self-adjoint differential equations,” Archives of ALGA, vol. 4, pp. 55–60, 2007. [13] N. H. Ibragimov, M. Torrisi, and R. Tracinà, “Quasi self-adjoint nonlinear wave equations,” Journal of Physics A, vol. 43, no. 44, Article ID 442001, 2010. [14] N. H. Ibragimov, M. Torrisi, and R. Tracinà, “Self-adjointness and conservation laws of a generalized Burgers equation,” Journal of Physics A, vol. 44, no. 14, Article ID 145201, 2011. [15] M. Torrisi and R. Tracinà, “Quasi self-adjointness of a class of third order nonlinear dispersive equations,” Nonlinear Analysis, vol. 14, no. 3, pp. 1496–1502, 2013. [16] M. L. Gandarias, “Weak self-adjoint differential equations,” Journal of Physics A, vol. 44, no. 26, Article ID 262001, 2011. [17] M. L. Gandarias, M. Redondo, and M. S. Bruzón, “Some weak self-adjoint Hamilton-Jacobi-Bellman equations arising in financial mathematics,” Nonlinear Analysis, vol. 13, no. 1, pp. 340– 347, 2012. [18] N. H. Ibragimov, “Nonlinear self-adjointness and conservation laws,” Journal of Physics A, vol. 44, Article ID 432002, 2011. [19] N. H. Ibragimov, “Nonlinear self-adjointness in constructing conservation laws,” Archives of ALGA, vol. 7/8, pp. 1–90, 2011. [20] I. L. Freire and J. C. S. Sampaio, “Nonlinear self-adjointness of a generalized fifth-order KdV equation,” Journal of Physics A, vol. 45, no. 3, Article ID 032001, 2012. [21] I. L. Freire, “New classes of nonlinearly self-adjoint evolution equations of third- and fifth-order,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 3, pp. 493–499, 2013.

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